Order of Operations

When simplifying an expression, it is very important to follow the order of operations. We often times refer to PEMDAS as a reminder of the order we should preform the operations. PEMDAS represents parentheses, exponents, multiplication, division, addition, and subtraction. Each operation is discussed below.

1. Parentheses
   

First, simplify everything that appears inside each set of parentheses in the expression. Take the contents inside the parentheses to be a new expression, and simplify that expression following the order of operations.

Note that if an expression is divided by another expression, such as:

\[{{{3 + 2 \cdot 5} \over {4 - {{5 \over 7}}}}}\]

You should consider the numerator to be in a set of parentheses, and the denominator to be in a set of parentheses:

\[{{{\left( {3 + 2 \cdot 5} \right)} \over {\left( {4 - {{5 \over 7}}} \right)}}}\]

2. Exponents

Evaluate all exponents within the expression. For instance, $$2^3$$ should be simplified to 8.

3. Multiplication and Division

Perform whichever operation comes first in the expression as you read it from left to right. For example, consider the expression:

$$3 \cdot 4/6 \cdot7$$

We first multiply 3 by 4 to get 12. Then, we divide 12 by 6 to get 2, and finally multiply 2 by 7 and get a final answer of 14.

4. Addition and Subtraction

Perform whichever operation comes first in the expression as you read it from left to right. For example, if you have the expression:

$$7 - 2 + 5 - 6$$

We first subtract 2 from 7 to get 5. Then, we add 5 to get 10 and finally subtract 6 to get a final answer of 4.

 

Example 1:

Simplify using the order of operations:

\[5 - 3(2 + 3 \cdot 4^2 )\]

• First, we solve what is inside the parentheses, which in this case is the expression:
• \[2 + 3 \cdot 4^2 \]
• Since there are no parentheses, we move on to exponents. There is an exponent on the 4, we replace $$4^2$$ with 16 and get:
• \[2 + 3 \cdot 16\]
• Next, we look for multiplication or division. Here, we have $$3 \cdot 16 = 48$$. We can replace the multiplication by 48 and get: $$2 + 48$$
• Now, we add $$2 + 48 = 50$$, and we are done with the inside of the parentheses!
• Don’t forget that all we have here is the answer to inside the parentheses, thus we still need to go back and simplify the rest of the problem. At this point, the expression is:
• \[5 - 3\left( {50} \right)\]
• Don’t make the mistake to subtract 5-3 first!! Remember that $$3\left( {50} \right)$$ means 3 multiplied by 50, and we do multiplication before subtraction. So, we get:
• \[5 - 150\]
• Finally, subtract:
• \[5- 150 =  - 145\]
• Our final answer is -145

Some practice problems to check your skills:

1. Perform the operations to simplify:  $$3\left( {4 + {\large{{16} \over 4}}} \right) - 6 \cdot 2^2 $$

Think about it for a moment and then access this link to view answer.

2. Perform the operations to simplify:  $${\Large{1 \over 4}}\left( {3 + 11 \cdot {\large\left( {{2 \over {22}}}\right)}}\right) - {\Large{{35} \over 7}}$$

Think about it for a moment and then access this link to view answer.

3. Perform the operations to simplify:   $$8(4+3-5)^2 / 9$$

Think about it for a moment and then access this link to view answer.

4. Perform the operations to simplify:   $${\Large{{3 + 7\left( {6 - 2} \right)} \over {7^2 }}} + 6 \cdot \left( { - 3 + 4} \right)$$

Think about it for a moment and then access this link to view answer.

5. Perform the operations to simplify:  $$7+56/8 \cdot 6+3-4 \cdot 3+1$$

Think about it for a moment and then access this link to view answer.